| L(s) = 1 | − 2·2-s − 2·3-s + 4·4-s − 9·5-s + 4·6-s − 31·7-s − 8·8-s − 23·9-s + 18·10-s + 57·11-s − 8·12-s − 52·13-s + 62·14-s + 18·15-s + 16·16-s + 69·17-s + 46·18-s + 19·19-s − 36·20-s + 62·21-s − 114·22-s − 72·23-s + 16·24-s − 44·25-s + 104·26-s + 100·27-s − 124·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.384·3-s + 1/2·4-s − 0.804·5-s + 0.272·6-s − 1.67·7-s − 0.353·8-s − 0.851·9-s + 0.569·10-s + 1.56·11-s − 0.192·12-s − 1.10·13-s + 1.18·14-s + 0.309·15-s + 1/4·16-s + 0.984·17-s + 0.602·18-s + 0.229·19-s − 0.402·20-s + 0.644·21-s − 1.10·22-s − 0.652·23-s + 0.136·24-s − 0.351·25-s + 0.784·26-s + 0.712·27-s − 0.836·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 19 | \( 1 - p T \) |
| good | 3 | \( 1 + 2 T + p^{3} T^{2} \) |
| 5 | \( 1 + 9 T + p^{3} T^{2} \) |
| 7 | \( 1 + 31 T + p^{3} T^{2} \) |
| 11 | \( 1 - 57 T + p^{3} T^{2} \) |
| 13 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 17 | \( 1 - 69 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 + 150 T + p^{3} T^{2} \) |
| 31 | \( 1 - 32 T + p^{3} T^{2} \) |
| 37 | \( 1 + 226 T + p^{3} T^{2} \) |
| 41 | \( 1 + 258 T + p^{3} T^{2} \) |
| 43 | \( 1 + 67 T + p^{3} T^{2} \) |
| 47 | \( 1 - 579 T + p^{3} T^{2} \) |
| 53 | \( 1 + 432 T + p^{3} T^{2} \) |
| 59 | \( 1 + 330 T + p^{3} T^{2} \) |
| 61 | \( 1 + 13 T + p^{3} T^{2} \) |
| 67 | \( 1 + 856 T + p^{3} T^{2} \) |
| 71 | \( 1 - 642 T + p^{3} T^{2} \) |
| 73 | \( 1 + 487 T + p^{3} T^{2} \) |
| 79 | \( 1 + 700 T + p^{3} T^{2} \) |
| 83 | \( 1 + 12 T + p^{3} T^{2} \) |
| 89 | \( 1 + 600 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1424 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.54940433550039781248905296178, −14.23296422964362277452297025381, −12.28910345705112146395900577343, −11.74116232900522629401248886174, −10.05869653671102961037539039729, −9.040103969382764092311668381850, −7.34057999284608864934117657958, −6.07862671895595820014035289771, −3.44697715328199501815487651605, 0,
3.44697715328199501815487651605, 6.07862671895595820014035289771, 7.34057999284608864934117657958, 9.040103969382764092311668381850, 10.05869653671102961037539039729, 11.74116232900522629401248886174, 12.28910345705112146395900577343, 14.23296422964362277452297025381, 15.54940433550039781248905296178
